Dalton
A transformation is the same change to all points. The viewing window is the part of the coordinate plain you see graphed. The Default window is the normal graphing window. You can make sure your graph is acurate by adjusting the window scale. All graphs should include labled axes, scaleon axes, an accurate representation of shape, include all intercepts, and points of discontinuety. And that is it!
In the lesson of 3-1 you are taught how to change windows on an automatic grapher "a.k.a an automatic grapher such as your TI-84 Plus Silver".
-Nate P.
This lesson talks about translation images, which are graphs under a translation. The graph before the translation is call the preimage. Translations are transformations which moves the graph up, down, left, or right using addition and subtraction. To apply a translation using points (x,y), map them onto (x+h,y+k). 'h' and 'k' are values that represent where the graph is going. When h is positive in x+h, the preimage will be translated to the right and if h is negative, it will go to the left. Same for k in y+k, if k is positive it goes up, if k is negative it goes down. Now heres a theorem that discribes this process. You can (1) replace x with x - h and y with y-k in a sentence; (2) apply the translation (x,y)--->(x+h,y+k) to the graph of the original relation. Ether work.
Summaries from Jeremy G and Lantz M?
A translation of a data set is a translation that maps one point onto another, and forms the translation image which is the new set of points. The measure of centers theorem states when you add a number to the numbers in the data set, you add that number to the mean, median, and mode. The measure of spread theorem states when you add each number in the data set you don't change the range,IQR or the standard deviation.
A figure is Reflection-Symmetric if and only if it can be mapped onto itself by a reflection over some line. The reflecting line L is called the axis or line of symmetry of the figure. A figure is 180'-rotation-symmetric or has or has symmetry to point P or point symmetry if and only if the figure can be mapped onto itself under a rotation of 180' around point P. The point P is called a center of symmetry. A graph is symmetric to the y-axis if for every point (x,y) on the graph, it's reflection image over the y-axis, (-x,y), is also on the graph. A graph is symmetric to the x-axis if for every point (x,y) on the graph, it's reflection image over the x-axis, (x,-y), is also on the graph. A graph is symmetric to the origin if and only if for every (x,y) on the graph, (-x,-y), is also on the graph. A power function is a function F with an equation of the form f(x)=x^nth power, when N is a positive integer greater than or equal to 2. A function F is an even function if and only if for all values of x in its domain, f(-x)=f(x). A function F is an odd function if and only if for all values of x in its domain, f(-x)=-f(x).
Mary G.
This lesson deals mostly with types of symmetry a graph can have. For example, reflection symmetric is just reflected over a line. There is also rotation symmetry where the figure is mapped onto itself by rotating it 180 degrees. This lesson also deals with functions of a graph. The functions can be odd or even. You can tell if the function is odd by plugging in a point for x so that f(-x)=-f(x). You can tell if it is an even function by plugging in a point for x so that f(-x)=f(-x). The function can also be neither odd or even, so you have to be careful.
Alyssa
A scale change is a transformation that stretches or shrinks a graph both vertically and horizontally. The value that changes the horizontal values of a graph is a horizontal scale factor and the value that changes the vertical values of a graph is a vertical scale factor. A size change occurs when the horizontal and vertical scale factors are the same. The graph scale-change theorem states that when dealing with a graph, the following transformations will give the same graphs: 1) Replacing x with x/a and y with y/b in the equation 2) applying the scale change (x,y)-> (ax,by) to the graph where a is the horizontal scale factor and b is the vertical scale change factor. When using a negative scale factor, the values will be reflected over an axis.
Austin B.
When a scale change is applied to the x-value then it will stretch or shrink horizontally, making it the horizontal scale factor. When a scale change is applied to the y-value the it will stretch or shrink vertically, making it the vertical scale factor. If the same scale change is applied to the x-value and the y-value then it is considered a size change and making the image similar to the preimage. If the scale change factor is negative then it will reflect over an axis. If the negative scale change is applied to the x-value then it will reflect over the y-axis (all the x-values will be made negative) but when the negative scale change is applied to the y-value then it will reflect over the x-axis (all the y-values will be made negative).
A scale change on a data set is a transformation which maps each element x onto ax, where a does not equal 0. The number (a) that you multiply each element in the data to cause a change is called the scale factor. After applying a scale change to a set of data, the point where the original data point ends up is called the scale image. Scaling or rescaling is when you apply a scale change to a data set. To find the measures of center of a scale image you just multiply each measure by the scale factor. The measures of spread are different. The variance of the scale image is the square of a times the original variance. The standard deviation and range are |a| times their original values.
-Dan H.
The book says that “a scale change of a set of data {x1, x2, ..., xn} is a transformation that maps each xi to axi, where a is a nonzero constant (a.k.a. a doesn’t equal 0).” That is, S is a scale change if and only if S:x –> ax, or S(x) = ax.
In this equation, a is the scale factor, which tells how much bigger or smaller the graph will become. When applied to the set of data, each number is multiplied by the scale factor, or a.
Also, the scale factor affects the median, mean, and mode by being multiplied by each of them. The way it affects the standard deviation and variance is that the variance is a2 times the original variance, and the standard deviation is |a| times the original standard deviation along with the range being changed by the same amount.
Composition of functions is mostly a review of a skill that we learned last year in Advanced Algebra. A composite of functions can be writen as f•g(x) or f(g(x)). Both notations mean the same thing. When solving a composite of functions you must work from the inside of the parenthesis out, just like solving a normal problem. A composite of functions is not commutative so f(g(x)) is not the same as g(f(x)).
-Katie S.
Compositions of functions are simple if one can understand the concepts. Clearly all of this is review from last year. Functions can be written out in two different ways, f(g(x)) or f•g(x). Both of these notations mean the same exact thing. It is extremely important to work from the inside out, and to remember that compositions are not commutative. f(g(x)) is not the same as g(f(x)). Also remember that the variable is just a place holder and stands for something else.
The domain is not always all real numbers, and is restricted to the first function in the composition, and one must examine the final domain.
-Troy L.
Inverse functions are functions that do the exact opposite of what another function has done. For example, if it were possible, let's say Fred, from his home, walks three miles west and happens to grow abnormally three times as tall when he gets to his location because of a pill he took before he left. The inverse of this situation would be Fred shrinking three times his height and walking three miles east. He is exactly the same as when he left. This is exactly what inverse functions do. This situation does pose a problem, however. Fred can walk home first and then shrink. In mathematics, this is not so. The last action Fred did in the original situation, grow three times as tall, relates to the first action he does in the inverse, shrink 3 times as small.
To find the inverse of a function, the x-values and y-values must be switched in the original function. This is especially easy in a function whose domain and range both have single points (and/or whose graph is discontinuous, sometimes). In an equation, this works just the same. In the original function, switch the x and y, or x and f(x), and solve for y or f(x). Although it's not mandatory, it's a good idea to replace f(x) with y. This helps to avoid confusion.
The Horizontal Line Test (HLT) is a test used to find if a graph's inverse is a function. In a graph, a horizontal line is drawn. If it hits two or more points, the graph's inverse is not a function. This does not mean that the graph itself is not a function.
The Inverse Function Theorem is a theorem which basically says that two functions are inverses if you can apply both to a variable (commonly "x") in any order and come out with x. In the Fred example, he could do the either scenario first. He could shrink three times his size and then walk three miles east and then walk three miles west and grow three times the size OR he could do vice versa. If one order of the functions work and the other doesn't, the inverse is not really an inverse. BOTH orders must work. Also, note- in either order in the Fred example, he ends up as same old Fred at home. This is the same for "x." If f and g are inverses of each other, then f(g(x))=x in the domain of g and g(f(x))=x in the domain of f. The same old "x" is left at the end.
-Robert M.
Inverse functions are all about "undoing" what another function already did. To "undo" a function, one would do the opposite of what they did to that origianl function. A basic example would be subtracting 5 from a number then multiplying it by 12. To get back to the original state, one would divide by 12 and then add five. Essentially it is where one switches the independent variable with the dependent variable. The notation of inverse f is to the -1 power. The theorem for the Horizontal-Line Test tells somebody if the inverse of the function is a function or not. This tells nothing about the original function. That is something just to think about.
A z-score represents how many standard deviations an individual point is away from the mean. An equation to find the z-score of data where z is the z-score, x is the data point, x bar is the mean, and s is the standard deviation is: z = x- x bar /s (Sorry, I can't get the equation typed as it normally would be written, but it is found on Page 216 in the book). A positive z-score indicates how far the data point is above the mean in terms of standard deviation, while a negative z-score tells how many standard deviations below the mean the data is. The original data is called raw data or raw scores. The results of the transformation, also known as the z-scores, are called standardized data or standardized scores. One theorem in Lesson 3-9 mentions that the mean of the z-scores of any data will always equal 0, and the standard deviation of the z-scores will always be 1. By using z-scores in real life situations, one can determine how his or her score or salary compares to a group as a whole.
Other than computing a percentile, another way of determining a score or salary relative to a group as a whole is by finding its z-score. The z-score of a number tells a person how many standard deviations (the average distance a set of data is away from its mean) away the number is above or below the mean of the data that it was in. A person can find a z-score of a number, salary, or wage compared to the others in its set by first subtracting the mean of the data from the number, salary, or wage (x), and then deviding the resulting number by the data's standard deviation (s). The formula " (x - mean) / s" is the general formula for calculating a z-score. Raw data or raw scores are the original data (the origan scores, salaries, or numbers of a data set you may want to transform into a z-score). The results of a transformation that turned the raw data(scores) into z-scores are called the standardized scores or standardized data. When the raw data is converted into standardized data, the mean of the data becomes zero. This is because all the z-scores of the data added together equal 0. Also, when this same transformation happens, the standard deviations of the standardized data becomes 1. The book describes this situation as follows in its theorem: If a data set has a mean "x-bar" and standard deviation "s", the mean of its z-scores will be zero, and the standard deviation of its z-scores will be one. Standardized scores, or z-scores, make it easier to compare different sets of numbers.
Table of Contents
Chapter 3: Transformations of Graphs and Data
3-1: Changing Windows
DaltonA transformation is the same change to all points. The viewing window is the part of the coordinate plain you see graphed. The Default window is the normal graphing window. You can make sure your graph is acurate by adjusting the window scale. All graphs should include labled axes, scaleon axes, an accurate representation of shape, include all intercepts, and points of discontinuety. And that is it!
In the lesson of 3-1 you are taught how to change windows on an automatic grapher "a.k.a an automatic grapher such as your TI-84 Plus Silver".
-Nate P.
TI Connect Video
3-2: The Graph-Translation Theorem
Andrew and ShelbyThis lesson talks about translation images, which are graphs under a translation. The graph before the translation is call the preimage. Translations are transformations which moves the graph up, down, left, or right using addition and subtraction. To apply a translation using points (x,y), map them onto (x+h,y+k). 'h' and 'k' are values that represent where the graph is going. When h is positive in x+h, the preimage will be translated to the right and if h is negative, it will go to the left. Same for k in y+k, if k is positive it goes up, if k is negative it goes down. Now heres a theorem that discribes this process. You can (1) replace x with x - h and y with y-k in a sentence; (2) apply the translation (x,y)--->(x+h,y+k) to the graph of the original relation. Ether work.
3-3: Translations of Data
Summaries from Jeremy G and Lantz M?
A translation of a data set is a translation that maps one point onto another, and forms the translation image which is the new set of points. The measure of centers theorem states when you add a number to the numbers in the data set, you add that number to the mean, median, and mode. The measure of spread theorem states when you add each number in the data set you don't change the range,IQR or the standard deviation.
3-4: Symmetries of Graphs
Nate M.A figure is Reflection-Symmetric if and only if it can be mapped onto itself by a reflection over some line. The reflecting line L is called the axis or line of symmetry of the figure. A figure is 180'-rotation-symmetric or has or has symmetry to point P or point symmetry if and only if the figure can be mapped onto itself under a rotation of 180' around point P. The point P is called a center of symmetry. A graph is symmetric to the y-axis if for every point (x,y) on the graph, it's reflection image over the y-axis, (-x,y), is also on the graph. A graph is symmetric to the x-axis if for every point (x,y) on the graph, it's reflection image over the x-axis, (x,-y), is also on the graph. A graph is symmetric to the origin if and only if for every (x,y) on the graph, (-x,-y), is also on the graph. A power function is a function F with an equation of the form f(x)=x^nth power, when N is a positive integer greater than or equal to 2. A function F is an even function if and only if for all values of x in its domain, f(-x)=f(x). A function F is an odd function if and only if for all values of x in its domain, f(-x)=-f(x).
Mary G.
This lesson deals mostly with types of symmetry a graph can have. For example, reflection symmetric is just reflected over a line. There is also rotation symmetry where the figure is mapped onto itself by rotating it 180 degrees. This lesson also deals with functions of a graph. The functions can be odd or even. You can tell if the function is odd by plugging in a point for x so that f(-x)=-f(x). You can tell if it is an even function by plugging in a point for x so that f(-x)=f(-x). The function can also be neither odd or even, so you have to be careful.
3-5: The Graph Scale-Change Theorem
AlyssaA scale change is a transformation that stretches or shrinks a graph both vertically and horizontally. The value that changes the horizontal values of a graph is a horizontal scale factor and the value that changes the vertical values of a graph is a vertical scale factor. A size change occurs when the horizontal and vertical scale factors are the same. The graph scale-change theorem states that when dealing with a graph, the following transformations will give the same graphs: 1) Replacing x with x/a and y with y/b in the equation 2) applying the scale change (x,y)-> (ax,by) to the graph where a is the horizontal scale factor and b is the vertical scale change factor. When using a negative scale factor, the values will be reflected over an axis.
Austin B.
When a scale change is applied to the x-value then it will stretch or shrink horizontally, making it the horizontal scale factor. When a scale change is applied to the y-value the it will stretch or shrink vertically, making it the vertical scale factor. If the same scale change is applied to the x-value and the y-value then it is considered a size change and making the image similar to the preimage. If the scale change factor is negative then it will reflect over an axis. If the negative scale change is applied to the x-value then it will reflect over the y-axis (all the x-values will be made negative) but when the negative scale change is applied to the y-value then it will reflect over the x-axis (all the y-values will be made negative).
3-6: Scale Changes of Data
A scale change on a data set is a transformation which maps each element x onto ax, where a does not equal 0. The number (a) that you multiply each element in the data to cause a change is called the scale factor. After applying a scale change to a set of data, the point where the original data point ends up is called the scale image. Scaling or rescaling is when you apply a scale change to a data set. To find the measures of center of a scale image you just multiply each measure by the scale factor. The measures of spread are different. The variance of the scale image is the square of a times the original variance. The standard deviation and range are |a| times their original values.-Dan H.
The book says that “a scale change of a set of data {x1, x2, ..., xn} is a transformation that maps each xi to axi, where a is a nonzero constant (a.k.a. a doesn’t equal 0).” That is, S is a scale change if and only if S:x –> ax, or S(x) = ax.
In this equation, a is the scale factor, which tells how much bigger or smaller the graph will become. When applied to the set of data, each number is multiplied by the scale factor, or a.
Also, the scale factor affects the median, mean, and mode by being multiplied by each of them. The way it affects the standard deviation and variance is that the variance is a2 times the original variance, and the standard deviation is |a| times the original standard deviation along with the range being changed by the same amount.
-Andrea M.
3-7: Composition of Functions
Composition of functions is mostly a review of a skill that we learned last year in Advanced Algebra. A composite of functions can be writen as f•g(x) or f(g(x)). Both notations mean the same thing. When solving a composite of functions you must work from the inside of the parenthesis out, just like solving a normal problem. A composite of functions is not commutative so f(g(x)) is not the same as g(f(x)).-Katie S.
Compositions of functions are simple if one can understand the concepts. Clearly all of this is review from last year. Functions can be written out in two different ways, f(g(x)) or f•g(x). Both of these notations mean the same exact thing. It is extremely important to work from the inside out, and to remember that compositions are not commutative. f(g(x)) is not the same as g(f(x)). Also remember that the variable is just a place holder and stands for something else.
The domain is not always all real numbers, and is restricted to the first function in the composition, and one must examine the final domain.
-Troy L.
3-8: Inverse Functions
Inverse functions are functions that do the exact opposite of what another function has done. For example, if it were possible, let's say Fred, from his home, walks three miles west and happens to grow abnormally three times as tall when he gets to his location because of a pill he took before he left. The inverse of this situation would be Fred shrinking three times his height and walking three miles east. He is exactly the same as when he left. This is exactly what inverse functions do. This situation does pose a problem, however. Fred can walk home first and then shrink. In mathematics, this is not so. The last action Fred did in the original situation, grow three times as tall, relates to the first action he does in the inverse, shrink 3 times as small.
To find the inverse of a function, the x-values and y-values must be switched in the original function. This is especially easy in a function whose domain and range both have single points (and/or whose graph is discontinuous, sometimes). In an equation, this works just the same. In the original function, switch the x and y, or x and f(x), and solve for y or f(x). Although it's not mandatory, it's a good idea to replace f(x) with y. This helps to avoid confusion.
The Horizontal Line Test (HLT) is a test used to find if a graph's inverse is a function. In a graph, a horizontal line is drawn. If it hits two or more points, the graph's inverse is not a function. This does not mean that the graph itself is not a function.
The Inverse Function Theorem is a theorem which basically says that two functions are inverses if you can apply both to a variable (commonly "x") in any order and come out with x. In the Fred example, he could do the either scenario first. He could shrink three times his size and then walk three miles east and then walk three miles west and grow three times the size OR he could do vice versa. If one order of the functions work and the other doesn't, the inverse is not really an inverse. BOTH orders must work. Also, note- in either order in the Fred example, he ends up as same old Fred at home. This is the same for "x." If f and g are inverses of each other, then f(g(x))=x in the domain of g and g(f(x))=x in the domain of f. The same old "x" is left at the end.
-Robert M.
Inverse functions are all about "undoing" what another function already did. To "undo" a function, one would do the opposite of what they did to that origianl function. A basic example would be subtracting 5 from a number then multiplying it by 12. To get back to the original state, one would divide by 12 and then add five. Essentially it is where one switches the independent variable with the dependent variable. The notation of inverse f is to the -1 power. The theorem for the Horizontal-Line Test tells somebody if the inverse of the function is a function or not. This tells nothing about the original function. That is something just to think about.
-Evan M.
3-9: z-Scores
A z-score represents how many standard deviations an individual point is away from the mean. An equation to find the z-score of data where z is the z-score, x is the data point, x bar is the mean, and s is the standard deviation is: z = x- x bar /s (Sorry, I can't get the equation typed as it normally would be written, but it is found on Page 216 in the book). A positive z-score indicates how far the data point is above the mean in terms of standard deviation, while a negative z-score tells how many standard deviations below the mean the data is. The original data is called raw data or raw scores. The results of the transformation, also known as the z-scores, are called standardized data or standardized scores. One theorem in Lesson 3-9 mentions that the mean of the z-scores of any data will always equal 0, and the standard deviation of the z-scores will always be 1. By using z-scores in real life situations, one can determine how his or her score or salary compares to a group as a whole.
----------------------------------------------------------------------------------------------------> Zach G.<--------------------------------------------------------------------------------------------------------
Other than computing a percentile, another way of determining a score or salary relative to a group as a whole is by finding its z-score. The z-score of a number tells a person how many standard deviations (the average distance a set of data is away from its mean) away the number is above or below the mean of the data that it was in. A person can find a z-score of a number, salary, or wage compared to the others in its set by first subtracting the mean of the data from the number, salary, or wage (x), and then deviding the resulting number by the data's standard deviation (s). The formula " (x - mean) / s" is the general formula for calculating a z-score. Raw data or raw scores are the original data (the origan scores, salaries, or numbers of a data set you may want to transform into a z-score). The results of a transformation that turned the raw data(scores) into z-scores are called the standardized scores or standardized data. When the raw data is converted into standardized data, the mean of the data becomes zero. This is because all the z-scores of the data added together equal 0. Also, when this same transformation happens, the standard deviations of the standardized data becomes 1. The book describes this situation as follows in its theorem: If a data set has a mean "x-bar" and standard deviation "s", the mean of its z-scores will be zero, and the standard deviation of its z-scores will be one. Standardized scores, or z-scores, make it easier to compare different sets of numbers.
By: Austin R.